**What is Mastery?**

The essential idea behind mastery is that all children need a deep understanding of the mathematics they are learning[1]. It is not just being able to memorise key facts and procedures and answer test questions accurately and quickly. It involves knowing ‘why’ as well as knowing ‘that’ and knowing ‘how’. It means being able to use one’s knowledge appropriately, flexibly and creatively and to apply it in new and unfamiliar situations[1].

Broadly, the mastery approach includes, but not limited to, the following:

**Intelligent Practice:** The arrangement of tasks and exercises to draw pupils’ attention to patterns, structure and mathematical relationships[2].

**Variation Theory:** What is same and what is different in a given set of tasks

**Multiple representations:** Pictorial, concrete and abstract representations of mathematical concepts

There is also an explicit focus on covering one topic in great depth and moving on to a new topic only when the child’s understanding is secure in the previous topic. The whole class ends up doing the same activity at the same time, with the pupils who need to be challenged given questions of greater depth on the same topic, rather than being given a different topic.

From a pupil’s point of view, a pupil really understands a mathematical concept, idea or technique if he or she can:

- describe it in his or her own words
- represent it in a variety of ways (e.g. using concrete materials, pictures and symbols – the CPA approach)
- explain it to someone else
- make up his or her own examples (and non-examples) of it
- see connections between it and other facts or ideas
- recognise it in new situations and contexts
- make use of it in various ways, including in new situations

Developing mastery with greater depth is characterised by pupils’ ability to:

- solve problems of greater complexity (i.e. where the approach is not immediately obvious), demonstrating creativity and imagination
- independently explore and investigate mathematical contexts and structures, communicate results clearly and systematically explain and generalise the mathematics.[1]

In the next post, I will discuss about what Active learning means. Meanwhile, I’m curious to know, what is mastery, from your perspective?

**References:**

- Teaching for Mastery: Questions, tasks and activities to support assessment, Mike Askew, Sarah Bishop, Clare Christie, Sarah Eaton, Pete Gri n and Debbie Morgan, https://www.ncetm.org.uk/public/files/23305581/Mastery_Assessment_Y3_Low_Res.pdf, 14/10/2015
- Intelligent Practice, http://www.mathshubs.org.uk/bespoke/april-2015/intelligent-practice/, 14/10/2015

*Megan wants to fill a bucket with water. A bucket holds 6 litres. A jug holds 500 millilitres. How many jugs of water does Megan need to fill an empty bucket?*

What exactly is going on here? Why is the student unable to make the link between the problem presented with the tools he already has at his disposal? Some of the common patterns I’ve observed so far are:

- The student tries to apply any technique that comes to his/her mind, without thinking through what exactly is being asked. When I ask the student why he/she chose an operation, more often than not, it turns out to be an “instinctive guess”
- The student wants to get to an answer as quickly as possible, and then check with me if he/she got it right. If not, he/she tries another strategy, and checks with me again – just trial and error, with not enough attention to the actual question, but focus on just getting “an” answer.
- The student is unable to understand the question, even after reading it multiple times.

While we can work with the first two scenarios mostly by reassuring the student that the process is more important than the answer, the third point is of particular interest to me: How good is a Maths intervention, when a child is still struggling with “Reading Comprehension” issues?

I recently learned that **92% of 11 to 14 year olds have a reading comprehension age of 8 or below**. This figure is beyond shocking, and no wonder students struggle with Maths, or any subjects that require a good understanding of the problem being posed.

As an adult, the material that I find difficult to comprehend are the ones that I’m not yet familiar with. When I chose a new subject area, I take my time to warm up to the terminologies and common knowledge in the domain, and then the subject matter starts making sense to me. I may be being naive here, but I don’t see how that can be different for a child.

Are we missing a trick by providing dry word problems that they can’t relate to, and hence have trouble in comprehending them? In other words, do they really care if Megan fills the bucket or not? What would make them care? Perhaps examples that are relevant to their daily life? On another note, what if relevant examples are not enough by themselves? What if there was a way for them to experience or act on these examples? What if Maths is situated so well in the context, that it’s no longer “Maths as a practice”, but becomes “Maths as an experience“? Is this possible, or practical?

I dare say, Yes! If Rob is at the front with a bucket and a jug, and if Emily is trying to figure out how many trips Rob is going to make from the water tap to the bucket with his jug, she may actually care. She may actually do the Math. If you have jugs of different sizes, and challenge Rob to find the minimum number of trips he would need to fill the bucket up, he might join in the fun as well. And at the end of the exercise, you may not have to ask the students how many jugs would Megan have needed. They know.

Am I right in thinking the word problems we pose the children may not be suited for their reading comprehension abilities/interests? Are we asking them the **right** questions?

**Organization:** I can imagine how overwhelming it must be to organize an event at such a grand scale (when I say grand, I mean GRAND). I am very thankful for i2i events group and Edtech UK for making it all look seamless with their effort and support.

**Exposure:** It is not often you get the opportunity to talk to hundreds of your users under the same roof. The feedback we got from talking to teachers, and watching the teachers and students play our game was invaluable.

**Market:** Just a walk down the aisle reminded us of the number of companies who are working on solving the same problems we are trying to solve, using various approaches. It was fascinating to look at where their products are at, and where we stand in comparison. A good reminder to sit down and execute faster and better.

**Pitch:** I admit, the pitch we had to do during the event was a bit nerve-wracking, having had little time to prepare. It went alright though, I suppose, as we then had people coming over and talking to us, referring to what we had pitched earlier on. Again, it was nice to be able to pitch in front of an audience that matters.

**Community:** For me, this is the biggest takeaway from BETT. Standing there, with 30 other fellow entrepreneurs who are passionate about what they do, and spending 4 days of our time reaching out to our users was an experience in itself. Little things, like referring a teacher to another startup if they don’t provide the service the teacher was looking for, exchanging marketing materials to place on each other’s desks, exchanging relevant leads etc – it was not as much a competitive world, but very much a collaborative one. Perhaps it’s the nature of the market we are in – educators sure do know the value of sharing!

Despite being thoroughly exhausting, we loved every minute of exhibiting at BETT, and we can’t wait to be back next year!

]]>This will serve as a good starting point for further more robust evaluations we intend to carry out this year in association with Dartington Social Research Unit. We very much look forward to sharing our progress with you. Happy 2016!

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Hope you find our work interesting. Please email me at prathap@scidice.co.uk to get in touch.

]]>When I did get to play players who are much better than me, although I enjoyed it at the beginning, very soon it was becoming a futile exercise. Apart from losing games, which can sometimes be demoralising, I also started becoming conscious of wasting the opponent’s time.

Then it stuck me – the implications challenges can have on learning. As you might agree, we need to present the right level of challenge for a child to stay motivated to solve a given problem. You could stretch a child to learn her 10 times table if she is comfortable with her 5’s. But if she is yet to master counting on, times tables can be too big a challenge, for example.

A lot has been said of adaptive learning and customising questions to a student’s ability, but not a lot has been spoken about the implications it can have on a child’s confidence. Continuing from the previous example, if the child is comfortable with her 5’s, but for some reason continues to struggle with her 10’s for a sustained period of time, would a worksheet with loads of questions of 10’s than 5’s be the right way to stretch the child’s ability? On the other hand, if we load the worksheet with 5’s because she’s struggling with her 10’s, are we doing enough to motivate her?

I don’t claim to have answers. I’m very curious to know if you do. For now, in our game Mathscraft, we have introduced zones of comfort:

- Comfort zone
- Stretch zone
- No-go zone

Going with the same example, her comfort zone would be her 5’s, stretch zone would be 10’s, and no-go zone would be the other times tables. We calculate the probability of she getting her answer right, based on previous answers, and we move her to stretch zone, only when the probability passes a defined level. We also automatically revert her back to her comfort zone, if the probability of she getting her answers right in her stretch zone goes below a defined level.

We are still experimenting with the definition of these levels, and we hope to find the optimal levels that will keep the child motivated. We think this is the best way to stretch a child’s ability – would you agree? I’m also curious to know if you have other thoughts/opinions on this – please email me if you do at prathap@mathscraft.com

]]>In today’s post, we are going to briefly discuss the **3 main reasons why Maths anxiety could occur**:

- Teachers’ anxiety
- Parents’ anxiety
- Teaching to test

**Teachers’ anxiety**

While this is probably not as common as the other two reasons, and most teachers, as a rule of thumb, overcome their anxiousness rather quickly early in their career, I just briefly wanted to mention this, before moving on to discuss more prevalent causes.

I am not a qualified teacher (I’m a volunteer tutor), so I can’t speak for the rest of the NQT’s, but I have plenty of teacher friends of mine who could identify with the kind of anxiousness issues I faced in my first few months in my classroom. In my case, my anxiousness did not necessarily stem from the lack of knowledge in the subject area, but from the lack of teaching toolset I needed to have, to teach them effectively. I’ve been a tutor for close to a year now, and I’m just beginning to learn the ropes. I’m lucky though, as I have an amazing class teacher who I work with, who makes up for my shortcomings.

As for my NQT friends, they are some of the smartest people I’ve known. But still, when it comes to actual classroom practice, their first year seems to stress them the most, as they go about figuring out how to apply the practices they have learnt in the classroom.

**Parents’ anxiety**

Adults who have not had access to the best maths education practices growing up, tend to be anxious about their maths skills, which could also affect their child’s affinity towards maths. In the UK, 4 in 5 adults have a low level of numeracy, according to a report from National Numeracy. A study conducted by the University of Chicago on 438 children aged 6-8 found that parents, who felt anxious about using maths but provided frequent help at home, slowed their child’s progress due to being less confident in explaining mathematical concepts.

**Teaching to test**

This, according to me, is probably the biggest factor contributing to maths anxiety amongst young children. If you’ve read Jo Boaler’s Elephant in the classroom, you might remember the monster the child drew when asked to draw about how she felt about maths.

I have observed this in my classroom as well – my students find assessments stressful, and barring the best performing children, almost every student thinks of it as a yardstick by which their self worth is measured. At such a young age, it’s hard for them to see their shortcomings as an opportunity to improve. This is where, I think, the teacher’s role becomes critical – in assuring the child that assessments are reflection of progress, not personality.

At our school, we try our best to encourage children to use their formative assessment scores as feedback tools, but it becomes a much harder sell when SATs are approaching. I can’t think of a easy way to address this – I’d love to hear your thoughts no this issue.

What do you think are the causes of Maths anxiety? And do you have any best practices I could learn from when it comes to keeping children motivated as SATs approach?

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Today, I want to talk about what Maths anxiety is, and what I have observed in my classroom.

Let’s agree what we mean by Maths anxiety first. It is defined as

*“a feeling of tension, apprehension, or fear that interferes with math performance” (2002, p. 1)[1]*

As maths teachers, many of you might already be able to relate to this. In my classroom, I have observed a range of emotions, right from complete disengagement, to feeling embarrassed, to feel angry, or upset, when my children engage with maths. When I try to dig deeper into these emotions, it is their fear and apprehension that stands out.

How could you know if it is indeed anxiety that a child is going through? The best indicator that is available is **avoidance**. When a child is actively trying to distract himself/herself, or prefers to engage in other activities whenever presented with a maths problem, it must ring you an alarm bell.

Of course, Maths anxiety is not a discrete feeling associated with Maths as a whole; it is an emotion experienced in various degrees of strength, depending on the competency of the child and the complexity of the problem the child is presented with. The biggest risk though, is that avoidance may result in decreased competency, and thus leading to further more anxiety, affecting the child’s confidence in Maths altogether.

There are so many reasons why this situation could occur, which I am going to reserve for the next post, before moving on to some of the ways we can handle it. For now, I’m keen on hearing your thoughts on this:

Have you observed Maths anxiety in your classroom? How did you conclude that it was anxiety?

**References:**

1. Ashcraft, M.H. (2002). Math anxiety: Personal, educational, and cognitive consequences.Directions in Psychological Science, 11, 181-185.

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Given that play doesn’t have any purpose other than the enjoyment of the act itself, what role could it have on learning? Before dwelling deeper on this topic, let’s agree on what learning is, at least in the context of this article. Among the various ways we can define learning, the one I am particularly in agreement with is this:

*“To acquire, or attempt to acquire knowledge or an ability to do something”. *

I guess the reason I like this definition is that it not only covers the act, but also the intent, which is to apply one’s knowledge obtained through learning.

Why is this definition relevant, and how is it connected to play? We will come back to it in a minute. But before that, let’s dig a bit deeper into play, shall we? So what do children enjoy in play? When I was a child (even now as a 31 year old), I enjoyed these aspects:

**Competition**: The challenge to stretch my abilities to accomplish something, and obtain a social validation for it**Collaboration**: The ability to play together with my peers to get something done as a team**Agency**: No one told me what to do. Win or otherwise, I chose my destiny

I’m sure there may be other factors I have enjoyed, but these are the ones that come to the top of my mind, and I hope you are able to identify with some of those too.

This is when it gets interesting: these aspects are very much part of learning too. Every child I know wants to be the best version of himself/herself (competition/identity), they love working with groups, especially when not constantly supervised by adults (collaboration), and they’d much rather know their strengths and weaknesses to make an effort accordingly (Agency). In fact, the AFL framework is based on the premise of the child’s agency in learning.

When play and learning seem so similar to each other, surely, our children must be playing a lot more in the classroom, aye? Well, it may not be that straight forward. Have you heard of “**Active learning**”? It’s a term coined in the Scottish curriculum for excellence. It stands for “Spontaneous and planned purposeful play” – essentially, the teacher scaffolding playful sessions facilitating learning.

While it may seem like a great idea, the reality is more complicated than that. Some of the factors that come into play are:

- High student to teacher ratio
- Behavioural issues in the classroom
- Time pressure on target outcomes

If you want to know more, here’s an excellent research on this topic done in Scotland.

Where I teach, our play sessions are quite successful, not least because we facilitate our sessions by:

- Telling our children what to expect during the session
- Telling them what I expect from them
- Ensuring every child gets something at the end of the session

What play sessions have you tried? And what worked and what didn’t? Please let me know through the comments here or email me; I’d love to know!

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